Are any matrices positive semidefinite, non-negative, and not diagonally dominated?

Question

If so, I'd appreciate any examples. Thanks.

Answer 1

A $2\times 2$ positive (symmetric) matrix $\begin{bmatrix}a&b\\b&d\end{bmatrix}$ is positive definite iff its determinant $ad-b^2$ is positive. If $a>b$, then it is very easy to choose $d < b$ such that $ad > b^2$, making it positive definite but not diagonally dominant.

Answer 2

For all $n\ge3$, the $n\times n$ matrix $J_n$ with all entries equal to $1$ is nonnegative, positive semidefinite ($x^\ast J_nx=|\sum_ix_i|^2\ge0$) and not diagonally dominant.